151

1 + 5 + 1

Digit sum repeated until one digit remains; equals n mod 9 (with 9 for multiples of 9).

Reads the same in both directions.

Leaves remainder 1 on division by 2.

Divisible only by 1 and itself.

1 distinct prime factor, 1 with multiplicity.

No prime divides it twice.

(−1) to the power of 1 distinct prime factors.

A gap of 8 between them.

1, 151

The proper divisors — everything except n itself.

Its proper divisors fall short by 150.

How many integers from 1 to n share no factor with n.

A prime congruent to 3 mod 4 divides it an odd number of times, which rules it out.

Excluded by Legendre: it has the form 4^a(8b + 7).

Lagrange's four-square theorem: every non-negative integer is, without exception.

Not triangular, square, pentagonal, hexagonal, cubic, Fibonacci, Lucas, Catalan, factorial or a power of two.

Summing the squares of its digits falls into the 4 → 16 → 37 → … → 4 cycle.

Not divisible by its digit sum 7.

Peaks at 1,024 along the way.